Estimation of Recovery Factor during a Waterflood Aristide Kamga Latalle Imperial College Supervisor: Dr. Ann Muggeridge Abstract The objective of this analysis is to assess the accuracy of the sweep efficiency formula, which is used in order to predict secondary recovery. The formula is defined as the product of the sweep efficiencies. The assessment was carried out by comparing theoretical predictions of reservoir performance with numerical simulation results. In order to investigate some of the parameters affecting reservoir performance, a significant number of conceptually modelled reservoirs with various rock and fluid properties were ana- lysed for the purpose of the study. The project has identified numerous contributions developed by researchers such as Stiles (1949), Dykstra & Parsons (1950) and many others over the years. The correlations developed for prediction of reservoir performance have shown that a reservoir pro- duction profile is affected by a large number of parameters, some of which are easily identified. Such parameters include: the oil- water viscosity ratio, the oil-water density difference, the geographical locations of the wells, the geology of the reservoir and many other rock and fluid properties, whose influences are recognised, but hardly quantified. At the end of the analysis, the fol- lowing conclusion was established: the actual performance of a waterflood rarely matches the theoretically predicted results. The main reason for the inconsistencies is the fact that it is virtually impossible and impractical to incorporate all the variables affecting the performance of a waterflood into a theoretical prediction technique. Some of the parameters causing the inconsisten- cies are: heterogeneity across the reservoir, gravitational cross flow between layers, well pattern across the reservoir. The analysis contained in this paper showed that the recovery factor for a waterflooded reservoir, can reasonably be predicted using Stiles’ and Dykstra-Parsons’ methods, assuming that the reservoir is very simple, in terms of geology, geometry and fluid properties. Alt- hough in the case of a model with complex heterogeneity, the waterflood performance is more difficult to predict theoretically. In this situation the use of numerical simulators can help in predicting future reservoir performances. Introduction Over the years, the petroleum industry has developed numerous methods of estimating primary and secondary recovery efficien- cies for hydrocarbon reservoirs. One of the most popular tools for improving recovery efficiencies is the injection of water into a reservoir, this is done in order to maintain the pressure inside the reservoir, and displace the hydrocarbons towards the production wells (Willhite, 1986). In order to determine the viability of a waterflood programme, reservoir engineers face the important task of forecasting what extra amount of hydrocarbons can be extracted from a field, as a result of the water injection (Craft, 1959). Earlier work on this subject focused mostly on the methods that are used to estimate primary recovery efficiencies (Iloabachie, 2009;Arps et al., 1967; Guthrie and Greenberger, 1955). Iloabachie (2009) also performed a preliminary evaluation of one of the methods used to evaluate secondary recovery efficiencies. One of the most established methods of analytically predicting secondary recovery efficiencies is by working out the product of three main parameters: the microscopic displacement efficiency (E D ), the areal sweep efficiency (E A ) and the vertical sweep effi- ciency (E V ) (Craig, 1971). - Buckley and Leverett (1942) described two-phase immiscible displacement in a linear system. Their work was further devel- oped by Welge (1952), who presented a graphical method of estimating the microscopic displacement efficiency (E D ), which is defined as the fraction of movable oil that has been displaced from the swept zone at any given time (Ahmed, 2006). - The areal sweep efficiency (E A ) is the fractional area of the pattern that is swept by the displacing fluid (Ahmed, 2006). Nu- merous authors such as Dyes et al. (1954) along with Kimbler et al. (1964) and Fassihi (1986) have presented graphical and analytical correlations for estimating the areal sweep efficiency. - The vertical sweep efficiency (E V ) is the fraction of the vertical section of the pay zone that is contacted by injected fluids (Ahmed, 2006). It can be evaluated by using two well-known methods; the first one was developed by Stiles (1949) and the second one was presented by Dykstra and Parsons (1950). - Dietz (1953) described an analytical methodology, which is used to estimate the recovery efficiency in a reservoir, where gravitational effects dominate the flow of fluids and affect vertical sweep. Thus it can be seen that the secondary recovery factor depends on a number of parameters, which includes; flood pattern, reser- voir heterogeneity (areal and vertical), oil and water viscosities (and densities). The geological information and especially the de- gree of heterogeneity are particularly very crucial when planning a water injection program (Dromgoole and Speers, 1997). Pre- Imperial College London
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Estimation of Recovery Factor during a Waterflood Aristide Kamga Latalle Imperial College Supervisor: Dr. Ann Muggeridge
Abstract The objective of this analysis is to assess the accuracy of the sweep efficiency formula, which is used in order to predict secondary
recovery. The formula is defined as the product of the sweep efficiencies. The assessment was carried out by comparing theoretical
predictions of reservoir performance with numerical simulation results. In order to investigate some of the parameters affecting
reservoir performance, a significant number of conceptually modelled reservoirs with various rock and fluid properties were ana-
lysed for the purpose of the study.
The project has identified numerous contributions developed by researchers such as Stiles (1949), Dykstra & Parsons (1950) and
many others over the years. The correlations developed for prediction of reservoir performance have shown that a reservoir pro-
duction profile is affected by a large number of parameters, some of which are easily identified. Such parameters include: the oil-
water viscosity ratio, the oil-water density difference, the geographical locations of the wells, the geology of the reservoir and
many other rock and fluid properties, whose influences are recognised, but hardly quantified. At the end of the analysis, the fol-
lowing conclusion was established: the actual performance of a waterflood rarely matches the theoretically predicted results.
The main reason for the inconsistencies is the fact that it is virtually impossible and impractical to incorporate all the variables
affecting the performance of a waterflood into a theoretical prediction technique. Some of the parameters causing the inconsisten-
cies are: heterogeneity across the reservoir, gravitational cross flow between layers, well pattern across the reservoir. The analysis
contained in this paper showed that the recovery factor for a waterflooded reservoir, can reasonably be predicted using Stiles’ and
Dykstra-Parsons’ methods, assuming that the reservoir is very simple, in terms of geology, geometry and fluid properties. Alt-
hough in the case of a model with complex heterogeneity, the waterflood performance is more difficult to predict theoretically. In
this situation the use of numerical simulators can help in predicting future reservoir performances.
Introduction Over the years, the petroleum industry has developed numerous methods of estimating primary and secondary recovery efficien-
cies for hydrocarbon reservoirs. One of the most popular tools for improving recovery efficiencies is the injection of water into a
reservoir, this is done in order to maintain the pressure inside the reservoir, and displace the hydrocarbons towards the production
wells (Willhite, 1986). In order to determine the viability of a waterflood programme, reservoir engineers face the important task
of forecasting what extra amount of hydrocarbons can be extracted from a field, as a result of the water injection (Craft, 1959).
Earlier work on this subject focused mostly on the methods that are used to estimate primary recovery efficiencies (Iloabachie,
2009;Arps et al., 1967; Guthrie and Greenberger, 1955). Iloabachie (2009) also performed a preliminary evaluation of one of the
methods used to evaluate secondary recovery efficiencies.
One of the most established methods of analytically predicting secondary recovery efficiencies is by working out the product of
three main parameters: the microscopic displacement efficiency (ED), the areal sweep efficiency (EA) and the vertical sweep effi-
ciency (EV) (Craig, 1971).
- Buckley and Leverett (1942) described two-phase immiscible displacement in a linear system. Their work was further devel-
oped by Welge (1952), who presented a graphical method of estimating the microscopic displacement efficiency (ED), which
is defined as the fraction of movable oil that has been displaced from the swept zone at any given time (Ahmed, 2006).
- The areal sweep efficiency (EA) is the fractional area of the pattern that is swept by the displacing fluid (Ahmed, 2006). Nu-
merous authors such as Dyes et al. (1954) along with Kimbler et al. (1964) and Fassihi (1986) have presented graphical and
analytical correlations for estimating the areal sweep efficiency.
- The vertical sweep efficiency (EV) is the fraction of the vertical section of the pay zone that is contacted by injected fluids
(Ahmed, 2006). It can be evaluated by using two well-known methods; the first one was developed by Stiles (1949) and the
second one was presented by Dykstra and Parsons (1950).
- Dietz (1953) described an analytical methodology, which is used to estimate the recovery efficiency in a reservoir, where
gravitational effects dominate the flow of fluids and affect vertical sweep.
Thus it can be seen that the secondary recovery factor depends on a number of parameters, which includes; flood pattern, reser-
voir heterogeneity (areal and vertical), oil and water viscosities (and densities). The geological information and especially the de-
gree of heterogeneity are particularly very crucial when planning a water injection program (Dromgoole and Speers, 1997). Pre-
Imperial College London
2 Estimation of Recovery Factor During a Waterflood
diction of waterflood recovery may also require the knowledge of the values of the residual oil saturation and injection efficiency
at the end of the flood (Vicente et al, 2001).
The main objectives of this study are; to assess the accuracy of the traditional method of estimating waterflood performance by
comparing theoretical estimates of recovery efficiencies with simulated predictions, and to make recommendations on the best
method of forecasting secondary recovery. During this study, a number of conceptual waterflooded models with different geolo-
gies and fluid properties, were built and simulated, and the results were compared with calculated recovery factors.
Model Data During this study, different types of conceptual models were considered. They were homogeneous, heterogeneous and layered
models with different geometries. Different well patterns were designed for the water injection schemes in each of the models and
the reservoir fluids were given different properties. A voidage replacement program was set up on some of the models, so that the
water injection rate would match the liquid production rate from those fields. These models were specifically designed as de-
scribed above in order to allow a thorough investigation into the impact of these parameters on the performance of a waterflood. In
total, over 175 models were built and simulated for the purpose of the study. A first set of 84 models (with dimensions of 3000ft ×
3000ft × 50ft and a size of 50 × 50 × 10 cells, giving a total of 25000 cells) were designed with no capillary transition zone
(MODEL I), while a second set of 84 models (with similar dimensions as MODEL I) were built with a large capillary transition
zone (MODEL II), and a third set of models (with dimensions of 1200ft × 2200ft × 100ft and a size of 60 × 220 × 5 cells, giving
a total of 66000 cells) were built by selecting a number of parameters from model 2 of the tenth SPE Comparative Solution Project
(MODEL III) (Christie & Blunt, 2001). Bigger grid sizes would have caused difficulties with memory allocation during the simu-
lations with the available 32-bit computers. Horizontal permeability distributions for some models in MODEL I & III can be
viewed in Figures 1 and 2, the rock and fluid properties are summarised in Tables 1a and 1b:
Table 1(a). Range of Rock and Fluid Properties Table 1(b). Relative Permeability data for MODEL I
APPENDICES B and C contain more models details. The set of MODEL I was rearranged in a number of series listed below:
Table 2. List of the models in MODEL I Figure 1. PermeabilityX in MODEL I Well Pattern
Δρ = Water-Oil Density Difference, lb/ft3; ρw = 64 lb/ft
3 and µw
= 1 cP
Figure 2. PermeabilityXY in MODEL III
Parameters Low Base High
Permeability x , mD 1 100 5000
Permeability y, mD 1 100 5000
Permeability z, mD 0.0 10 2000
Porosity, fraction 0.0 0.2 0.400
Initial water saturation, fraction 0.2 0.24 0.24
Reservoir thickness, ft 50 50 100
Initial reservoir pressure, psi 2500 2500 8000
Maximum water cut, fraction 0.95 0.95 0.95
Oil density, lb/ft3 40 54.70 54.70
Oil viscosity, cP 1 2 10
Oil formation volume factor, rb/stb 1.01 1.05 1.1
Sw Krw Kro
0.24 0.01 1
0.3 0.024 0.719
0.4 0.076 0.388
0.5 0.187 0.187
0.6 0.388 0.076
0.65 0.535 0.044
0.7 0.719 0.024
0.75 0.948 0.011
0.78 1 0.007
3 Estimation of Recovery Factor During a Waterflood
Estimating Waterflood Recovery Factors Using Different Sweep Efficiencies Microscopic Displacement Efficiency (Ahmed, 2006)
By assuming there is no gas inside the reservoir at the start of the water injection, and that the oil formation volume factor is con-
stant over the field life, the microscopic displacement efficiency after water has broken through can be expressed as:
ED =
wi
wiw
S
SS
1 (1)
Similarly, in the case of water breakthrough, it is expressed as:
EDBT =
wi
wiwBT
S
SS
1 (2)
Where Swi = Initial Water Saturation, Fraction
wS = Average Water Saturation at fw = 0.95, Fraction
fw = Reservoir water fractional flow, Fraction
wBTS = Average Water Saturation at Breakthrough, Fraction
EDBT = Microscopic Displacement Efficiency at Breakthrough, Fraction
The average water saturations before and after breakthrough are determined through the use of the Buckley-Leverett analysis,
which is carried out on the reservoir fluid saturation profiles (see APPENDIX D from Page 42) (Buckley & Leverett, 1942).
Areal Sweep Efficiency Defined as the fraction of the total flood pattern that is contacted by the injection fluid, the areal sweep efficiency depends mainly
on the following parameters (Ahmed, 2006):
1. Mobility ratio M
2. Flood pattern
3. Cumulative water injected Winj
1. The mobility ratio is defined as the mobility of the displacing fluid to the mobility of the displaced fluid. In the case of
water injection, the mobility ratio from the start of the flood to breakthrough is expressed as (Muskat, 1946):
MBT =
w
o
wiro
wBTrw
Sk
Sk
@
@ (3)
After water breakthrough has been reached, it is expressed as:
M =
w
o
wiro
wrw
Sk
Sk
@
@ (Muskat, 1946) (4)
Where wBTrw Sk @ = Relative Permeability of Water at wBTS
wrw Sk @ = Relative Permeability of Water at wS
wiro Sk @ = Relative Permeability of Oil at wiS
wo , = Viscosity of Oil and Water, respectively
2. The flood pattern is defined as the geometric arrangement, formed by injection and production wells, so that symmetrical
and interconnected networks are created. During this study, a number of well patterns were considered, they are:
Quarter five-spot and five-spot
Quarter nine-spot and nine-spot
Direct line drive
3. The cumulative water injected is the total volume of water injected into the reservoir as a result of the waterflood.
The areal sweep is affected by reservoir heterogeneity, Pitts and Crawford (1970) demonstrated that low areal sweep efficiencies
are to be expected when waterflooding heterogeneous reservoirs. They had established this fact by developing a relationship be-
tween the areal sweep efficiency and the permeability ratio inside a field.
Over the years, a number of methods have been developed to predict the areal sweep efficiency, they are split into the following
three phases of a waterflooding program:
Before breakthrough
At breakthrough
After breakthrough
4 Estimation of Recovery Factor During a Waterflood
Areal Sweep Efficiency at Breakthrough (EABT)
Craig (1955) suggested a graphical relationship that displays a link between the areal sweep efficiency at breakthrough and the
mobility ratio in a five spot pattern configuration. Willhite (1986) proposed the following mathematical equation, which is an ap-
proximation of the graphical relationship developed by Craig (1955):
EABT = 0.54602036 +
BTM
03170817.0 +
BTMe
30222997.0 - 0.00509693MBT (5)
Mortada and Nabor (1961) suggested another expression for the areal sweep efficiency at breakthrough;
EABT = dah
qtb
2 (6)
Where q = Flow Rate, bbl/d [m3/s]
tb = Time to Breakthrough, days [s]
d = Distance between rows of unlike wells, ft [m] a = Distance between rows of like wells, ft [m]
h = Reservoir Thickness, ft [m]
= Porosity, Fraction
Areal Sweep Efficiency after Breakthrough (EA)
Dyes et al. (1954) presented a graphical relationship that relates the areal sweep efficiency with the reservoir water cut fw and the
reciprocal of the mobility ratio 1/M in a five-spot well pattern configuration. They also presented a similar graph in the case of a
direct line drive well pattern. In the case of a nine-spot configuration, Kimbler et al.(1964) suggested a graphical relationship that
relates the areal sweep efficiency with the reservoir water cut and the mobility ratio M.
Details about how the areal sweep efficiencies at and after breakthroughs were estimated for MODEL I, can be found in AP-
PENDIX D (from Page 43).
Vertical Sweep Efficiency Vertical sweep efficiency depends mainly on the mobility ratio and total volume injected (Ahmed, 2006). The injection fluid will
tend to move through the reservoir with an irregular front, as a result of the reservoir vertical heterogeneity. The degree of permea-
bility variation is therefore a key parameter influencing the vertical sweep efficiency (Ahmed, 2006). One of the most widely used
descriptor of vertical heterogeneity for a reservoir formation is the Dykstra-Parsons permeability variation coefficient V.
The Dykstra-Parsons Permeability Variation Coefficient V
Dykstra and Parsons (1950) initially presented the concept of the permeability variation coefficient V, which is a statistical meas-
ure of non-uniformity of a set of data, and is used to quantitatively describe the degree of heterogeneity within a reservoir. The
steps for determining the coefficient V can be found in APPENDIX D (from Page 45).
Two methods are commonly used to estimate the vertical sweep efficiency EV in a layered reservoir, they include: Stiles’ method
and the Dykstra-Parsons’ method. A layer is defined as a single thickness of one or more rock(s) with similar properties.
Stiles’ Method
Stiles (1949) suggested that the vertical sweep efficiency in a layered reservoir can be estimated by using the following formula:
EV =
n
j
j
i
n
ij
j
i
j
j
i
hk
khhk
1
11
(7)
Where i = Breakthrough Layer, i.e., i = 1,2,3, … n
n = Total number of layers
hi = Layer Thickness, ft [m]
ht = Total Thickness, ft [m]
Stiles also proposed the following formula for determining the surface water-oil ratio (WORs) as breakthrough occurs in any layer:
WORs = A
n
ij
j
i
j
j
kh
kh
1
1 (8)
5 Estimation of Recovery Factor During a Waterflood
With A =
ro
rw
k
k
ww
oo
B
B
(9)
Where rwk = Relative permeability to water at orS
rok = Relative permeability to oil at wiS
Sor = Residual oil saturation, Fraction
Equations (7), (8) and (9) are used simultaneously to describe the sequential breakthrough as it occurs in layer 1 through layer n
(Ahmed, 2006). The values of the vertical sweep efficiency at and after breakthrough can then be estimated by using the plots of
vertical sweep efficiency (EV) against surface water-oil ratio (WORs).
The value of EVBT is determined by extrapolating the plot to the line of WORs = 0, while the value of EV is found by reading off
the value of the vertical sweep efficiency for a WORs = 19 stb/stb (which is determined from the 95% water cut limit set on the
models during the simulations).
The Dykstra-Parsons’ Method
De Souza and Brigham (1981) grouped the vertical sweep efficiency curves for 0 ≤ M ≤ 10 and 0.3 ≤ V ≤ 0.8 into one curve. A
combination of WORr, V, and M was used by the authors of the formula to define the correlation parameter Y (Ahmed, 2006):
Y = x
r
VM
VWOR
10137.18094.0
499.2948.184.0
(10)
With
x = 1.6453V2 + 0.935V – 0.6891
Where WORr = Reservoir Water-Oil Ratio, bbl/bbl The vertical sweep efficiency (EV) is then expressed as:
EV = a1 + a2ln(Y) + a3[ln(Y)]2 + a4[ln(Y)]
3 +a5/ln(Y) + a6Y (11)
Where the coefficients a1 through a6 are:
a1 = 0.19862608 a2 = 0.18147754
a3 = 0.01609715 a4 = - 4.6226385*10-3
a5 = -4.2968246*104 a6 = 2.7688363*10
-4
Equations (10) and (11) are also used to determine the vertical sweep efficiency at breakthrough, by setting WOR r = 0 bbl/bbl
and by using the values of the mobility ratio at breakthrough (MBT).
Volumetric Sweep Efficiency
The volumetric sweep efficiency EVol is another important sweep efficiency to be mentioned. It is the product of EA and EV at any
time during the flood (Ahmed, 2006):
EVol = EAEV (12)
The waterflood recovery factor (WFRF) can then be expressed as:
WFRF = EDEvol after breakthrough (13)
And WFRFBT = EDBTEvolBT at breakthrough (14)
The volumetric sweep efficiency is defined as the fraction of the reservoir that has been swept or not swept by the injected water.
It is an indication of the amount of additional oil recovery that exists in the unswept portion of the reservoir. It is expressed as:
Evol = orwi
gi
orwi
o
SS
SN
SSVP
BP
11. (15)
With Np = EA( wS - Swi)PV after breakthrough (16)
And NPBT = EABT( wBTS - Swi)PV at breakthrough (17)
Where Sgi = Gas saturation at the start of the waterflood, Fraction = 0
wBTS = Average water saturation at breakthrough, Fraction (from the Buckley-Leverett analysis)
NP = Cumulative oil produced after breakthrough, stb
6 Estimation of Recovery Factor During a Waterflood
NpBT = Cumulative oil produced at breakthrough, stb
Swi = Water saturation at the start of the waterflood, Fraction
P.V = Pore Volume, bbls
Craig (1971) suggested the following formulas for the volumetric sweep efficiency and the waterflood recovery factor:
12 Estimation of Recovery Factor During a Waterflood
The results shown in Figures 8, 9 and 10 seem to suggest that the techniques involving Stiles’ and Dykstra-Parsons’ methods
generate better performance predictions for a waterflood than other prediction methods using sweep efficiencies. In Figures 8(a),
8(b) and 8(c), a careful view of the charts show that the simulation results were better matched by the estimations from theories 1
and 2. In Figures 9 and 10, the line representing a perfect match between the simulation results and the theoretical results is de-
fined by the equation y = x and a linear correlation coefficient of R2 = 1. Looking at Figures 9 and 10, it appears that the predic-
tion methods that have generated trendlines, close to the line of perfect match, are again the ones involving Stiles’ and Dykstra-
Parsons equations. This is why these two methods were selected to predict waterflood performances for the highly heterogeneous
models of MODEL III, 3D Pictures of these models with different well patterns are shown in Figures 11.
Table 7. Recovery Factor Prediction Methods Compared for the models in MODEL III
y = 0.993x - 0.0157 R² = 0.8095
y = 0.8816x + 0.0648 R² = 0.7673
y = 0.7282x + 0.1521 R² = 0.3568
y = 1.2651x - 0.3831 R² = 0.6389
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
RF
(th
eo
reti
cal)
RF (simulation)
Theory 1
Theory 2
Theory 3
Theory 4
y = 0.8595x + 0.0103 R² = 0.9056
y = 0.9213x - 0.024 R² = 0.8363
y = 0.5852x + 0.0971 R² = 0.4589
y = 1.2447x + 0.09 R² = 0.7405
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1 0.2 0.3 0.4 0.5 0.6
RF
bt
(th
eo
reti
cal)
RF bt (simulation)
Theory 1
Theory 2
Theory 3
Theory 4
Well Pattern
Model With Gravity w
o
RF (simulation) % Difference between the Results
At fw= 0.95
Theory 1 Theory 2
At fw= 0.95
At fw= 0.95
5-spot
NO 1 0.569 -12 -29.1
NO 2 0.515 -15 -27.1
NO 10 0.313 -8 -7.34
YES 1 0.568 -11.9 -29
YES 2 0.516 -15.5 -27.3
9-spot
NO
10
0.240
24.6 25.4
Direct Line Drive
NO
10
0.365
-17.2 -16.7
Figure 10. Comparing Recov-ery Factors at Breakthrough
for all the models in MODEL I.
Figure 9. Comparing Recovery Factors after Breakthrough (at fw = 0.95) for all the mod-els in MODEL I.
13 Estimation of Recovery Factor During a Waterflood
The results displayed in Table 7 show a higher degree of inconsistency between the simulation results and the theoretical predic-
tions. It shows that the theoretical results obtained for the simple models (MODEL I) were more consistent with the simulation
results than the results obtained for the complex models (MODEL III). The methods involving the use of Stiles’ and Dykstra-
Parsons’ equations (i.e. Theories 1 & 2) produced a better result than the other methods using the product of the sweep efficiencies
to estimate secondary recovery. This could be explained by the fact that whilst it is practically impossible to factor all the numer-
ous parameters affecting waterflood performance, into one single analytical prediction methodology, these methods have come
very close to achieving that goal as discovered during the course of this study. They are able to take into account the influence of
the oil-water viscosity ratio (Buckley & Leverett, 1942), the effect of well pattern (Dyes et al., 1954; Kimbler et al., 1964), the
impact of reservoir geology (Stiles, 1949; Dykstra & Parsons, 1950) and the consequences of the oil-water density difference
(Dietz, 1953). In the case of a more complex reservoir, both methods have limitations. This is due to a number of reasons which
include; the fact that in a highly heterogeneous reservoir and due to very large amount of geological data available, a great deal of
computations have to be averaged to simplify calculations. As a result, the accuracy of the final result is likely to be significantly
affected, causing at times, large inconsistencies between the results obtained from Theories 1 and 2, and the simulation results.
Discussion Many parameters influence the ultimate fraction of a reservoir to be swept by an injected fluid; they include sand continuity, per-
meability variations, oil-water viscosity ratios, regular well patterns and gravitational crossflows (Dyes et al., 1953).
In fluid systems with low oil-water viscosity ratios, recoveries tend to be higher regardless of the reservoir geology; this might be
due to the fact that fingering did not occur during the waterflood of the models. The viscosity ratio directly affects oil mobility, as
well as the fractional flow profile and the areal sweep. Higher oil-water viscosity ratios lead to higher mobility ratios, which tend
to allow water-rich regions created by gravity segregation at the bottom of the more permeable medium to be spread over (Gaucher
and Lindley, 1959).
During the course of this project, it has become clear that, the fraction of a reservoir which will be swept by an injected fluid is a
function of the geometric arrangement of the production and injection wells (Ahmed, 2006). The location of the wells across a
reservoir mainly affects the areal sweep efficiency across the reservoir. Numerous methods have been developed over the years to
quantify the effect of the locations of the wells on the areal sweepout; some of these methods are empirical and are mainly repre-
sented by graphical correlations between the areal sweep, the fractional water cut and the mobility ratio. This study has shown one
crucial fact about the areal sweep efficiency though; it is significantly affected by the level of heterogeneity across a reservoir.
Determining the areal sweep out in a highly heterogeneous model is a complex problem, which has not yet been solved analytical-
ly, the use of a numerical simulator is one of the few ways to estimate the areal sweep in a highly heterogeneous reservoir. This
fact may explain some of the inconsistencies observed between the simulation and theoretical results generated during this study.
Heterogeneity is such an important parameter, which needs to be well understood, if a waterflooding operation is to succeed (Gu-
lick & McCain, 1998). Waterflooding performance in heterogeneous reservoirs sometimes depends upon the classification of grain
size distribution along the vertical direction. In a highly heterogeneous reservoir influenced by gravity, waterflood recovery de-
creases as the permeability variation increases. This decrease will be greater in a reservoir with decreasing upward permeability
trends, as the reservoirs achieve uniformity; the difference in performance “due to permeability trend” will reduce. Heterogeneity
may take place in both horizontal and vertical directions. During the study, it was identified that, one aspect of vertical heterogene-
ity is the permeability variation. It results from numerous geological processes that took place during the sedimentation of the res-
ervoir (Permadi et al., 2004). Pitts and Crawford (1970) carried out a study on the influence of the heterogeneity on areal sweep
efficiencies; they discovered that, the areal sweep depends greatly on the permeability ratio inside a reservoir.
The effect of gravity inside a reservoir may be quite significant on the performance of a waterflood. Fluid cross flow, due to vis-
cous and capillary forces, can significantly affect sweep efficiency in heterogeneous reservoirs (Fitzmorris et al., 1992). Reservoir
models with dip angle of 0o like the models considered in this study have the effect of density difference between water and oil,
neglected. However, with sufficient vertical permeability present in a heterogeneous reservoir, the advancing water in a high per-
meability layer tends to cross flow to the underlying oil zone in a low permeability layer due to the density difference between oil
and water. As investigated during this project, the waterflooding performance in the case of models with gravity may be different
from the performance of models without gravity (El-Khatib, 2003).
This analysis has also covered a number of methods of estimating secondary recovery, and which involves the use of the parame-
ter known as the volumetric sweep efficiency. Two methods using volumetric sweeps in their calculations were described by Ah-
med (2006) and Craig (1971) (i.e. Theories 3 & 4). Both methods did not produce good results for estimation of secondary recov-
ery efficiencies and a review of Figures 8(a), (b) and (c), is a clear indication of their poor predictions. As a result, these methods
could not be relied upon to predict secondary recovery, even for simple geological reservoirs. Their shortcomings are mainly due
to the fact that, they only take account of the influence of a small number of parameters, which affects waterflood recovery, thus
completely neglecting the effect of important parameters such as the oil-water density difference, the reservoir geology and many
others, which are more difficult to quantify such as areal sweep in a heterogeneous formation.
During the course of this study, the waterflood prediction methods using Stiles’ and Dykstra-Parsons’ methods produced ac-
ceptable results for simple models. However, when it comes to more complex or realistic models, these methods are not as reliable.
In this case, using a numerical simulator would be advised before making an important economic decision.
14 Estimation of Recovery Factor During a Waterflood
Summary and Conclusions During this project, a number of prediction methods for waterflood performance were investigated and compared with simulation
results, which were assumed to match actual reservoir performance. These comparisons were made by considering simple and
complex models, in terms of reservoir geology. A careful description of the models analysed in the study can be found in APPEN-
DICES B and C. The results showed that the future performance of a reservoir of simple geology is easier to predict than the future
performance of a complex heterogeneous formation. This means that, the reliability of the theoretical methods is subject to the
level of heterogeneity of the reservoir, they could be used to predict the secondary recovery efficiency of a very simple geological
model. However, when it comes to predicting secondary recovery for more complex heterogeneous models, it is wise for the engi-
neer to rely on the simulation predictions, rather than on the theoretical forecasts. As a third alternative, both theoretical and simu-
lation calculations could be used to predict waterflood recovery efficiency, this would allow the engineer to feel more confident
about the final results he will obtain from using both approaches.
The following conclusions were drawn from this analysis:
1. Areal sweep efficiency is significantly affected by complex heterogeneities, and due to the fact that there are very few an-
alytical methods of estimating it across complex reservoirs, a numerical simulator may be a more reliable way of evaluat-
ing it.
2. Waterflood theoretical prediction methods produce better results for simple models than for more complex models.
3. Stiles’ and Dykstra-Parsons’ methods produce acceptable results when it comes to predicting waterflood performance for
simple geological reservoirs.
4. When dealing with a highly heterogeneous formation, it could be wise for the engineer to use both theoretical and simula-
tion methods.
5. In order to reduce uncertainty, simulation results need to be used along with theoretical results before making an im-
portant decision.
Recommendations for Future Work 1. Using a streamline simulator in order to better understand the effect of the different sweep efficiencies across the models.
2. Finding another way of estimating the areal sweep efficiency across heterogeneous and layered models by considering the
permeability ratio across the field.
3. Investigating the effect of the length of the completion zone on the recovery factor and how it could be used to maximise
the efficiency.
4. Finding a way of predicting recovery in a reservoir with pressure communicating layers and gravitational crossflow.
5. Investigating how to predict waterflood recovery in a reservoir containing a capillary transition zone.
-
(a) (b)
(c) (d)
15 Estimation of Recovery Factor During a Waterflood
Nomenclature 2D = 2-Dimensional
3D = 3-Dimensional
a = Distance between rows of like wells, ft [m]
B = Formation Volume Factor, rb/stb
d = Distance between rows of unlike wells, ft [m] ED = Microscopic Displacement Efficiency, Fraction
EDBT = Microscopic Displacement Efficiency at Breakthrough, Fraction
EA = Areal Sweep Efficiency, Fraction
EABT = Areal Sweep Efficiency at Breakthrough, Fraction
EV = Vertical Sweep Efficiency, Fraction
EVBT = Vertical Sweep Efficiency at Breakthrough, Fraction
EVOL = Volumetric Sweep Efficiency, Fraction
EVOLBT = Volumetric Sweep Efficiency at breakthrough, Fraction
h = Reservoir Thickness, ft [m]
hi = Layer Thickness, ft [m]
ht = Total Thickness, ft [m]
i = Breakthrough Layer, i.e., i = 1,2,3, … n
k = Layer Permeability, md [m2]
kro = Relative Permeability to oil, Dimensionless
krw = Relative Permeability to water, Dimensionless
M = Mobility Ratio, Dimensionless
MBT = Mobility Ratio at Breakthrough, Dimensionless
n = Total number of layers
Np = Cumulative Oil Produced at any time during the Flood, stb [sm3]
NpBT = Cumulative Oil Produced at Breakthrough, stb [sm3]
P.V. = Pore Volume, bbls [rm3]
q = Flow Rate, bbl/d [m3/s]
RF = Recovery Factor, Fraction
RFBT = Recovery Factor at Breakthrough, Fraction
Sgi = Gas Saturation at the Start of the Flood, Fraction
Sor = Residual Oil Saturation, Fraction
Swi = Initial Water Saturation, Fraction
wS = Average Water Saturation, Fraction
wBTS = Average Water Saturation at Breakthrough, Fraction
tb = Time to Breakthrough, days [s]
V = Permeability Variation Coefficient, Dimensionless
Winj = Cumulative Water Injected, bbl/day [m3/s]
WFRF = Waterflood Recovery Factor After Breakthrough, Fraction
WFRFBT = Waterflood Recovery Factor At Breakthrough, Fraction
Figure 11. Water Saturation Distributions at the end of field life for some models in MODEL III defined with: (a) A five-spot well pattern and a viscosity ratio of 10 (b) A nine-spot well pattern and a viscosity ratio of 10 (c) A direct line drive pattern and a viscosity ratio of 10 (d) A five-spot well pattern and a viscosity ratio of 1
16 Estimation of Recovery Factor During a Waterflood
Subscripts
o = Oil
g = Gas
w = Water
i = Initial
BT = Breakthrough
References Ahmed, T.: Principles of Waterflooding. Reservoir Engineering Handbook, Third Edition, Elsevier, (2006) 932 – 1082.
Buckley, S., and Leverett, M., “Mechanism of Fluid Displacement in Sands,” Trans. AIME, 1942, Vol. 146, p.107-116.
Christie, M. A. and Blunt, M. J.: “Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques”, SPE Paper
72469 (2001).
Craft, B. C., And Hawkins, M., Applied Petroleum Reservoir Engineering. Englewood Cliffs, NJ: Prentice Hall, 1959.
Craig, F. F.: The Reservoir Engineering Aspects of Waterflooding, Henry L. Doherty Memorial Fund of AIME, Dallas, TX. (1971)
Vol. 3 126.
Craig, F., Geffen, T., and Morse, R., “Oil Recovery Performance of Pattern Gas or Water Injection Operations from Model Tests,”
JPT, Jan. 1955, pp. 7 – 15, Trans. AIME, P. 204.
Dake, L.,: Fundamentals of Reservoir Engineering, Elsevier, Amsterdam, (1978) 349.
De Souza, A., and Brigham, W., “A Study on Dykstra-Parsons Curves”, TR29. Palo Alto, CA: Stanford University Petroleum
Research Institute, 1981.
Dietz, D. N., “A Theoretical Approach to the Problem of Encroaching and By-Passing Edge Water.” Akad. Van Wetenschappen, 1953,
Amsterdam. Proc. V. 56-B, pp. 83.
Dromgoole, P., and Speers, R.: Geoscore: A method for quantifying uncertainty in field reserve estimates. Petroleum Geosciences, (1997).
Dyes, A., Caudle, B., and Erickson, R., “Oil Production After Breakthrough as Influenced by Mobility Ratio,” JPT, April 1954,
pp. 27 – 32; Trans. AIME, p. 201.
Dykstra, H., and Parsons, R., “The Prediction of Oil Recovery by Water Flood”, in Secondary Recovery of Oil in the United States,
2nd ed. Washington, DC: American Petroleum Institute, 1950, pp. 160 – 174.
Eclipse (E 100) Reference Manual, 2009.
Fassihi, M., “New Correlations for Calculation of Vertical Coverage and Areal Sweep Efficiency,” SPHERE, Nov. 1986, pp. 604-606.
Fitzmorris R. E., Kelsey F. J. and Pande K. K., “Effect of Crossflow on Sweep Efficiency in Water/Oil Displacements in
Heterogeneous Reservoirs”, SPE Paper 24901 (1992).
Gaucher, D. H. and Lindley, D. C., “Waterflood Performance in a Stratified , Five-Spot Reservoir – A Scaled-Model Study”, SPE
Paper 1311- G (1959).
Gulick, K. E. and McCain, W. D.:”Waterflooding Heterogeneous Reservoirs: An Overview of Industry Experiences and Practices”
SPE Paper 40044 (1998).
Kimbler, O. K., Caudle, B. H., and Cooper, H. E., JR.: “Areal Sweepout Behavior in a Nine-Spot Injection Pattern”, Trans., AIME
(1964) 231, PP. 199 – 202.
Mortada, M., Nabor, G. W., “An Approximate Method for Determining Areal Sweep Efficiency and Flow Capacity in Formations
With Anisotropic Permeability”, SPE Paper 16 (1961).
Muskat, M., Flow of Homogeneous Fluids Through Porous Systems. Ann Arbor, MI: J. W. Edwards, 1946.
Noaman A.F. El-Khatib, “Effect of Gravity on Waterflooding Performance of Stratified Reservoirs”, SPE
Paper 81465 (2003).
Permadi A. K., Yuwono I. P. and Simanjuntak A. J. S., “Effects of Vertical Heterogeneity on Waterflood Performance in Stratified
Reservoirs: A Case Study in Bangko Field, Indonesia”, SPE Paper 87016 (2004).
Petrel 2009, Schlumberger Petrel Manual, 2008.
Pitts, Gerald N., Crawford, Paul B., “Low Areal Sweep Efficiencies in Flooding Heterogeneous Rock”, SPE Paper 2866-MS
(1970).
Stiles, W., “Use of Permeability Distribution in Waterflood Calculations”, Trans. AIME, 1949, Vol. 186, p. 9.
Vicente, M., Crosta, D., Eliseche, L., Scolari, J., and Castelo, R.: “Determination of Volumetric Sweep Efficiency in Barrancas
Unit, Barrancas Field (2001).
Welge, H., “A Simplified Method for Computing Oil Recovery By Gas or Water Drive,” Trans. AIME, 1952, pp. 91-98.
Willhite, G. P., Waterflooding. Dallas: Society of Petroleum Engineers, 1986.
SI Metric Conversion Factors bbl × 1.589873* E-01 = m
3
cP × 1.0* E+00 = mPa.s
ft × 3.048* E-01 = m
mD × 0.987* E-15 = m2
lb/ft3 × 1.6018* E+01 = kg/m
3
psi × 6.89475* E+03 = Pa
17 Estimation of Recovery Factor During a Waterflood
APPENDICES
18 Estimation of Recovery Factor During a Waterflood
APPENDIX A: Literature Survey MILESTONES IN RECOVERY FACTOR ESTIMATION STUDY
TABLE OF CONTENTS
Table A – 1: List of milestones in the study of recovery factor estimation during a waterflood SPE
Paper No. Year Title Authors Contribution
933219 - G 1933 “The Mechanics of Porous Flow Applied to Water-
flooding Problems”
R.D. Wyckoff, H.G.
Botset And M. Muskat
The paper highlights the influence of numerous parameters on the perfor-
mance of a waterflood program.
934062 1934 “A Theoretical Analysis of Water-flooding Networks”
M. Muskat, R. D.
Wyckoff
The paper studies the problem of sim-ultaneous movement of water and oil
in connected sand.
942107 1942 “Mechanism of Fluid Displacement in Sands”
Buckley, S. E. and
Leverett, M. C.
The paper describes a flow equation, which is designed to determine the
water saturation profile in the reservoir at any given time during water injec-
tion.
949009 1949 “Use of Permeability Distribution in Water Flood
Calculations”
Stiles, W. E.
The paper presents a mathematical
derivation of the effect of permeability distribution in water flood performance.
API 1950 “The Prediction of Oil Recovery by Water Flood” Dykstra, H., and
Parsons, R.
This paper describes a number of analytical derivations used in order to determine the vertical sweep efficien-
cy.
124 - G
1952
“A Simplified Method for Computing Oil Recovery by Gas or Water Drive”
Henry J. Welge
This paper presents the mathematical derivations behind the graphical meth-od used to determine the microscopic
displacement efficiency.
309 - G
1953
“Oil Production After Breakthrough – As Influenced by Mobility Ratio”
A. B. Dyes, B. H. Caudle, R. A. Er-
ickson
This paper highlights the influence of well pattern on the areal sweep effi-
ciency.
16 1961 “An Approximate Method for Determining Areal
Sweep Efficiency and Flow Capacity in Formations with Anisotropic Permeability”
M. Mortada, G.
W. Nabor
This paper provides a simple method for determining the areal sweep effi-ciency for a formation in which the
permeability in the bedding plane is anisotropic.
184 1964 “Areal Sweepout Behavior in a Nine-Spot Injection
Pattern”
Kimbler, O. K., Caudle, B. H., and Cooper, H. E.,jr.
This paper presents a number of graphical relationships used to deter-mine the areal sweep efficiency in a
nine-spot injection pattern.
2004 - PA
1967
“Combination Method for Predicting Waterflood Performance for Five-Spot Patterns in Stratified
Reservoirs”
James A. Wasson, Leo A. Schrider
This paper highlights the effect of the viscosity ratio on the waterflood oil
recovery.
2866 - MS 1970
“Low Areal Sweep Efficiencies in Flooding Hetero-
geneous Rocks
Pitts, Gerald N., Crawford, Paul B.
This paper describes how reservoir heterogeneity affects areal sweep
efficiency.
17289 –MS
1988
“Prediction of Waterflood Performance in Stratified
Reservoirs”
Tompang, R.,
Petronas; Kelkar, B.G., U. of Tulsa
This study analyses the crossflow between layers in a reservoir and how it can affect waterflood performance.
68806
2001
“Determination of Volumetric Sweep Efficiency in
Barrancas Unit, Barrancas Field”
M. Vicente, D. Crosta, L. Eliseche,
J. Scolari and R. Castelo
This paper describes a number of methods used to determine the volu-
metric sweep efficiency.
19 Estimation of Recovery Factor During a Waterflood
SPE 933219 - G (1933) The Mechanics of Porous Flow Applied to Water-flooding Problems
Authors: R.D. Wyckoff, H.G. Botset And M. Muskat
Contribution to the understanding of waterflood recovery factor estimation:
The paper highlights the influence of numerous parameters on the performance of a waterflood program.
Objective of the paper:
To determine the performance of a waterflood scheme, by using analytical theories and the flooding of a physical model, purposely
built for the experiment.
Methodology used:
A model was built by using an electrolyte containing an ion indicator, then photographs were taken at various stages of the flood
and the results were analysed in the study.
Conclusions reached:
1. The effect of impermeable barriers on the shape of the flood is in connection with encroachment of flooding phenomena.
2. The gravity has an effect on the shape of the waterfront, in an homogeneous medium, the water edge is expected to make
its first appearance at the bottom of the wells.
3. The presence of gas masses will have a considerable effect upon the apparent permeability of the medium to liquid flow.
4. In the case that, the gas content is very high, the apparent permeability will become a small fraction of the permeability in
the absence of the gas.
Comments:
In order to predict the water displacement, a complete solution to the pressure distribution and also the streamlines is required.
20 Estimation of Recovery Factor During a Waterflood
SPE 934062 - G (1934) A Theoretical Analysis of Water-flooding Networks
Authors: M. Muskat, R. D. Wyckoff
Contribution to the understanding of waterflood recovery factor estimation:
The paper studies the problem of simultaneous movement of water and oil in a connected sand.
Objective of the paper:
To develop empirical methods of estimating the recovery factor of a waterflooded reservoir.
Methodology used:
A model was built for the experiment, the water was injected into the oil sand with the intention of displacing the oil and removing
it through prearranged output wells.
Conclusions reached:
1. The efficiency of a flooding network and the shape of the advancing flood depends on the geometry of the model.
2. The well spacing and arrangement has an effect on waterflood recovery.
3. From the general comparison between a theoretical example and results of a practical field experience , that well spacing
and arrangement are probably of relatively minor importance in determining the success of a flooding program.
4. Little recoverable oil is left behind after a natural water drive, whereas in a gas drive field as much oil recoverable by im-
proved methods may remain in the sand as has been produced by natural recovery methods.
Comments:
Steady state conditions are assumed for all experiences carried out during the study. The systems are assumed to be two dimen-
sional and the difference in viscosity between the water and oil is neglected.
21 Estimation of Recovery Factor During a Waterflood
SPE 309 - G (1953) Oil Production After Breakthrough – As Influenced by Mobility Ratio
Authors: A. B. Dyes, B. H. Caudle, R. A. Erickson
Contribution to the understanding of waterflood recovery factor estimation:
The paper explains the influence of flood patterns on areal sweep efficiencies.
Objective of the paper:
To highlight the link between the different parameters of mobility ratios, well patterns and areal sweep efficiencies after water
breakthrough has occurred.
Methodology used:
A number of small models were built from ¼-in. thick alundum plates, a water injection system was set up on the models. During
the injection process, x-ray images of the models are taken using a shadowgraph. And the areal expansion of the injected fluid is
recorded and analysed.
Conclusions reached:
1. The inclusion of the production after breakthrough which results from a continual enlargement of the sweepout pattern is
essential in making a comparison between different plans of operation.
2. The data presented in this paper can be combined with observed water flood behavior to give the engineer a means of
gaining knowledge concerning the magnitude of oil flow behind an invasion front and knowledge of the possible influ-
ence of permeability stratification on the flood.
Comments:
This paper focuses on five-spot and direct line drive patterns, the other well patterns were not considered during the study.
22 Estimation of Recovery Factor During a Waterflood
SPE 1472 - G (1960) Injection Rates – The Effect of Mobility Ratio, Area Swept, and Pattern
Author: John C. Deppe
Contribution to the understanding of waterflood recovery factor estimation:
Not much because the paper is focused on the effect of mobility ratio and areal sweep efficiency on the water injection rates.
Objective of the paper:
To calculate water injection rates after breakthrough by determining the areal sweep efficiency at breakthrough.
Methodology used:
Used a method of calculating injectivity by approximating the flood pattern with radial flow elements (or a combination of radial-
and linear-flow elements for some patterns such as the direct line drive).
Conclusions reached:
1. When the mobility of the fluid injected into a reservoir is different from the mobility of the reservoir fluid, the injectivity
changes rapidly during the early part of the operation – and then only gradually until breakthrough occurs.
2. Injectivity can be estimated after breakthrough by assuming that the flood front can be approximated by simple shapes.
3. The approximate equations of injectivity can be applied to irregular interior patterns and to boundary patterns.
Comments:
This paper assumes a mobility ratio of one for all calculations, and states that an estimate of breakthrough sweep efficiency is re-
quired to calculate injectivity after breakthrough.
23 Estimation of Recovery Factor During a Waterflood
SPE 16 (1961) An Approximate Method for Determining Areal Sweep Efficiency and Flow Capacity in Formations with Anisotropic Permeabil-
ity
Authors: M. Mortada, G. W. Nabor
Contribution to the understanding of waterflood recovery factor estimation:
Although it does not specify a formula for estimating waterflood recovery factor, it does however provide an equation to estimate a
parameter used in secondary recovery factor calculations: the areal sweep efficiency.
Objective of the paper:
To estimate the areal sweep efficiency at breakthrough in a formation with anisotropic permeability.
Methodology used:
Used 2D dimensional flow equations in order to determine the areal sweep efficiency at breakthrough and the flow capacity.
Conclusions reached:
1. The method of analysis provides a technique for estimating the areal sweep efficiency at breakthrough and the flow ca-
pacity associated with flooding a formation with anisotropic permeability.
2. In anisotropic formations, rows of wells should be oriented along the major axis of permeability as nearly as possible to
obtain larger sweep.
Comments:
This paper assumed two flood patterns, the staggered and the skewed line drive.
24 Estimation of Recovery Factor During a Waterflood
SPE 1359 (1965) Unit Mobility Ratio Displacement Calculations for Pattern Floods in Homogeneous Medium
Author: Hubert J. Morel-Seytoux
Contribution to the understanding of waterflood recovery factor estimation:
It specifies a formula for estimating waterflood recovery factor at breakthrough, it also provides equations to estimate two parame-
ters used in secondary recovery factor calculations: the displacement and areal sweep efficiencies at breakthrough.
Objective of the paper:
To estimate the displacement and areal sweep efficiencies at breakthrough in an homogeneous formation , with different flood
patterns.
Methodology used:
Derived pressure equations in order to determine the areal sweep efficiency at breakthrough as a function of the pattern geometry.
Conclusions reached:
1. Due to drastic assumptions in the displacement process, it is not expected that results presented in this paper would apply
accurately to actual pattern flooding.
2. The primary value of the present results lies in their use as part of more elaborate prediction procedures that do account
for mobility ratio, two-phase flow.
3. The very concise and simple form of all the results give new hope that exact solutions may also be obtained for non-unit
mobility ratio.
Comments:
This paper assumed a mobility ratio of one for all the equations that were developed.
25 Estimation of Recovery Factor During a Waterflood
SPE 2004 - PA (1967) Combination Method for Predicting Waterflood Performance for Five-Spot Patterns in Stratified Reservoirs
Authors: James A. Wasson, Leo A. Schrider
Contribution to the understanding of waterflood recovery factor estimation:
This paper suggests a method that combines a number of different methods, in order to evaluate the performance of a waterflood
program with a five-spot flood pattern.
Objective of the paper:
To develop an analytical method for predicting waterflood recovery efficiencies for a five-spot flood pattern.
Methodology used:
A number of relationships developed by Buckley and Leverett were developed in order to determine if an oil bank will form in
front of the flood front. Then, a method developed by Craig et al is used to determine the areal sweep efficiency, knowing the mo-
bility ratio.
Conclusions reached:
1. A method of incorporating several well-known analytical methods for predicting waterflood performance into one com-
posite method has been described.
2. This method has led to the elimination of some of the weaker assumptions, which are part of the prediction procedure.
Comments:
The paper assumes a five-spot flood pattern for the entire prediction procedure developed. It assumes a constant injection pressure
during the field life. There is no vertical cross-flow between permeable layers.
26 Estimation of Recovery Factor During a Waterflood
SPE 2866 - MS (1970) Low Areal Sweep Efficiencies in Flooding Heterogeneous Rock
Authors: Pitts, Gerald N., Crawford, Paul B.
Contribution to the understanding of waterflood recovery factor estimation:
The paper explains the reasons why the areal sweep efficiencies are lower in heterogeneous rocks, although it does not provide a
mathematical equation that can quantify the effect of the rock heterogeneity on the value of the areal sweep efficiencies.
Objective of the paper:
To investigate why the areal sweep efficiencies obtained when waterflooding a heterogeneous rock are lower than the values ob-
tained in homogeneous rocks.
Methodology used:
Simulated a 20*20 grid of a heterogeneous reservoir with the help of a streamline simulator.
Conclusions reached:
1. The areal sweep efficiency depends greatly on the permeability ratio.
2. Areal sweeps for very heterogeneous five-spot patterns were reduced to about one-third of the sweep expected in homo-
geneous media.
3. Heterogeneous rock systems provide a meandering set of streamlines resulting in extremely low areal sweep efficiencies
in some cases.
Comments:
The paper assumed that a reservoir pattern such as a five-spot pattern, direct line drive square or staggered-line drive pattern could
be represented by a matrix of several hundred rock blocks with different permeabilities.
27 Estimation of Recovery Factor During a Waterflood
SPE 24901 (1992) Effect of Crossflow on Sweep Efficiency in Water/Oil Displacements in Heterogeneous Reservoirs
Authors: R. E. Fitzmorris, F. J. Kelsey, K. K. Pande
Contribution to the understanding of waterflood recovery factor estimation:
This paper explains the importance of capillary crossflow as a recovery mechanism, in addition to viscous and gravity crossflow,
and how reservoir heterogeneity can affect hydrocarbon recovery.
Objective of the paper:
To determine the effect of vertical crossflow in a layered reservoir
Methodology used:
The paper examined the relative magnitudes of capillary, viscous and gravity crossflow in water-oil displacements in a two-
dimensional heterogeneous permeability field. The authors used geostatistical methods to generate the porosity and permeability
variations between the layers.
Conclusion reached:
1. Viscous and capillary crossflow in water and oil wet reservoirs can seriously affect oil recovery in a heterogeneous for-
mation.
2. Heterogeneities in the porosity and permeability distribution have a significant effect on waterflooding performance.
Comments:
The J-function was used in order to develop the analytical equations needed to link the formation geology with the hydrocarbon
recovery efficiency.
28 Estimation of Recovery Factor During a Waterflood
SPE 40044 (1998) Waterflooding Heterogeneous Reservoirs: An Overview of Industry Experiences and Practices
Authors: Karl E. Gulick, William D. McCain
Contribution to the understanding of waterflood recovery factor estimation:
Although it does not provide equations for calculating waterflood recovery factor. It does however explain how reservoir hetero-
geneity can affect waterflooding performance.
Objective of the paper:
To explain the effect of reservoir geology and how it can affect waterflooding performance.
Methodology used:
The paper reviewed a number of waterflood management practices, highlighted in key industry papers.
Conclusion reached:
Successful implementation and operation of a waterflood project, even in complex heterogeneous formations, is a matter of execut-
ing a well-conceived comprehensive plan, none of the elements of which are “rocket science”.
Comments:
The paper addresses operating philosophy, well spacing, pattern development, completions, injection water, and surveillance. Alt-
hough these factors are presented from a west Texas perspective, they are applicable to reservoirs having a high degree of vertical
and areal heterogeneity.
29 Estimation of Recovery Factor During a Waterflood
SPE 68806 (2001) Determination of Volumetric Sweep Efficiency in Barrancas Unit, Barrancas Field
Authors: M. Vicente, D. Crosta, L. Eliseche, J. Scolari and R. Castelo
Contribution to the understanding of waterflood recovery factor estimation:
Although it does not specify a formula for estimating waterflood recovery factor, it does however provide equations to estimate
parameters used in secondary recovery factor calculations: the displacement and volumetric sweep efficiency.
Objective of the paper:
To develop formulas for estimating the displacement and volumetric sweep efficiency in a formation.
Methodology used:
The paper used an analysis of the evaluation and surveillance of the waterflooding in the South of the Barrancas Field, after more
than 30 years of water injection.
Conclusions reached:
1. The estimation of the volumetric sweep efficiency and also the values of the remaining oil saturation, the displace-
ment efficiency and the injection efficiency were useful to know the potential of the field in order to recover addi-
tional oil. The management and actions to be carried out were evaluated on the basis of these values.
2. Fractional curve based on production data is a good and complementary tool to obtain reservoir parameters as the av-
erage water saturation in the water swept portion of the reservoir.
3. The evaluation of the volumetric sweep efficiency needs an accurate knowledge of the floodable pore volume of the
reservoir.
Comments:
The method used to estimate the volumetric sweep efficiency needs production and injection data, and can be applied to any con-
figuration of patterns.
30 Estimation of Recovery Factor During a Waterflood
SPE 81465 - MS (2003) Effect of Gravity on Waterflooding Performance of Stratified Reservoirs
Authors: Noaman A.F. El-Khatib, King Saud University
Contribution to the understanding of waterflood recovery factor estimation:
The study has developed an analytical solution for estimating waterflood recovery factor of stratified reservoirs with gravitational
effects between adjacent layers.
Objective of the paper:
To explain the effect of mobility ratio, gravity number and the Dykstra-Parsons coefficient of permeability variation on water-
flooding performance.
Methodology used:
The effect of gravity on waterflooding performance of stratified reservoirs was investigated experimentally and by means of nu-
merical reservoir simulators
Conclusion reached:
1. The study shows that gravitational crossflow delays water breakthrough in high permeability layers, increases oil recov-
ery and decreases water cut.
2. The effect of gravitational crossflow on the waterflood performance is more evident for the cases of unfavorable mobility
ratios and for cases of highly heterogeneous reservoirs.
3. The order of layer permeability distribution in the reservoir has a large effect on the waterflooding performance of strati-
fied reservoirs with gravity effects.
Comments:
For the models, with no vertical crossflow, the study assumes a piston like displacement in the different layers.
For the models, with vertical crossflow, the study assumes instantaneous crossflow between layers to keep the pressure gradient
the same in all layers at any distance.
31 Estimation of Recovery Factor During a Waterflood
APPENDIX B: PVT PROPERTIES MODEL I : SIMPLE MODELS WITH NO TRANSITION ZONE
Reference Pressure : 2500 psia Rock Compressibility : 3*10
-6 psi
-1 Oil-Water Contact : 10000 ft
Datum Depth : 4500 ft
Table B – 1. Water Properties
Bw (rb/stb) 1.0
Cw (psi-1) 3*10-6
μw (cp) 1.0
Table B – 2. Oil Properties
P (psi) Bo (rb/stb) μo (cp)
1000 1.1 1 2 5 10
2500 1.05 1 2 5 10
Table B – 3. Relative Permeability data
Table B – 4. Water, Gas and Oil densities
ρo 40 lb/ft3
640.74 kg/m3
ρo 54.697 lb/ft3
876.16 kg/m3
ρw 64 lb/ft3
1025.18 kg/m3
ρo 0.044 lb/ft3
0.705 kg/m3
Sw Krw Kro
0.24 0.009944675 1
0.3 0.024278992 0.719677278
0.4 0.076733604 0.38846387
0.5 0.1873379 0.1873379
0.6 0.38846387 0.076733604
0.65 0.535055777 0.04497983
0.7 0.719677278 0.024278992
0.75 0.94839812 0.011708619
0.78 1 0.007021604
32 Estimation of Recovery Factor During a Waterflood
MODEL II : Simple Models with a large transition zone in the reservoir
Reference Pressure : 2500 psia Rock Compressibility : 3*10
-6 psi
-1 Oil-Water Contact : 4550 ft
Datum Depth : 4500 ft
Table B – 5. Water Properties
Bw (rb/stb) 1.0
Cw (psi-1) 3*10-6
μw (cp) 1.0
Table B – 6. Oil Properties
P (psi) Bo (rb/stb) μo (cp)
1000 1.1 1 2 5 10
2500 1.05 1 2 5 10
Table B – 7. Relative Permeability data
Table B – 8. Water, Gas and Oil densities
ρo 45.27 lb/ft3
725.15 kg/m3
ρo 54.697 lb/ft3
876.16 kg/m3
ρw 64 lb/ft3
1025.18 kg/m3
ρg 0.044 lb/ft3 0.705 kg/m
3
Sw Krw Kro Pcow (psi)
0.24 0.009944675 1 2.486
0.3 0.024278992 0.719677278 2.35
0.4 0.076733604 0.38846387 2.15
0.5 0.1873379 0.1873379 1.8
0.6 0.38846387 0.076733604 1.53
0.65 0.535055777 0.04497983 1.34
0.7 0.719677278 0.024278992 1.2
0.75 0.94839812 0.011708619 1.05
0.78 1 0.007021604 0.9
33 Estimation of Recovery Factor During a Waterflood
MODEL III : MODELS WITH COMPLEX HETEROGENEITY
Reference Pressure : 8000 psia Rock Compressibility : 3*10-6
psi-1
Oil-Water Contact : 12500 ft Datum Depth : 12000 ft
Table B – 9. Water Properties
Bw (rb/stb) 1.01
Cw (psi-1) 3*10-6
μw (cp) 0.3, 1.5, 3.0
Table B – 10. Oil Properties
P (psi) Bo (rb/stb) μo (cp)
300 1.05 2.85
800 1.02 2.99
8000 1.01 3.0
Table B – 11. Relative Permeability data
Sw Krw Kro
0.2 0 1
0.25 0.006944444 0.840277778
0.3 0.027777778 0.694444444
0.35 0.0625 0.5625
0.4 0.111111111 0.444444444
0.45 0.173611111 0.340277778
0.5 0.25 0.25
0.55 0.340277778 0.173611111
0.6 0.444444444 0.111111111
0.65 0.5625 0.0625
0.7 0.694444444 0.027777778
0.75 0.840277778 0.006944444
0.8 1 0
Table B – 12. Water and Oil densities
ρo 53 lb/ft3
848.98 kg/m3
ρo 40 lb/ft3
640.74 kg/m3
ρw 64 lb/ft3
1025.18 kg/m3
34 Estimation of Recovery Factor During a Waterflood
APPENDIX C : Rock Properties and Dimensions for the Models
MODEL I : SIMPLE MODELS WITH NO TRANSITION ZONE A number of different types of models were simulated during this study, they are homogeneous, heterogene-
ous and layered models. They were investigated with different flood patterns. Their data are tabulated in the
following tables:
Reference pressure: 2500 psia Datum depth: 4500 ft
Dimensions Grid Kx (md) Ky (md) Kz (md) Porosity Wells
3000ft*300ft*10ft 50*1*2 1000 1000 10
0.2
1 producer 1 injector
3000ft*300ft*10ft 50*1*2 200 200 1
3000ft*300ft*10ft 50*1*2 100 100 10
3000ft*300ft*10ft 50*1*2 50 50 1
3000ft*300ft*10ft 50*1*2 10 10 0.1
INJECTOR
PRODUCER
Figure C - 4. 2D vertical cross-section model
38 Estimation of Recovery Factor During a Waterflood
MODEL II : Simple Models with a large transition zone These models have a very large transition zone , as defined in APPENDIX B. Their data are tabulated in the following tables:
Reference pressure: 2500 psia Datum depth: 4500 ft
- QUARTER FIVE-SPOT PATTERN MODELS
Table C – 13. Homogeneous models – ρo = 54.697 lb/ft3 and ρo = 45.27 lb/ft
3
Dimensions Grid Kx (md) Ky (md) Kz (md) Porosity Wells
3000ft*3000ft*50ft
50*50*10 100 100 10 0.2 1 producer 1 injector
Table C – 14. Heterogeneous models – ρo = 54.697 lb/ft3 and ρo = 45.27 lb/ft
3
Dimensions Grid Kx (md) Ky (md) Kz (md) Porosity Wells
2. Determining the areal sweep efficiency at and after breakthrough in the five-spot pattern The areal sweep efficiency at breakthrough in a five-spot is determined, either by using Figure D-3 or by using Equation (5). And
the areal sweep efficiency after breakthrough in a five-spot is determined by using Figure D-4, at a water cut of 0.95.
Table D - 3. Areal Sweep Efficiency at and after Breakthrough in the five-spot pattern model µo
(cp) MBT M EABT EA
1 0.60 0.95 0.761 1
2 0.88 1.68 0.703 0.97
5 1.40 2.25 0.636 0.95
10 1.50 4 0.627 0.91
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
fw
Sw
1
2
5
10
Viscosity Ratio
Sw(BT)
44 Estimation of Recovery Factor During a Waterflood
Figure D-3. Areal Sweep Efficiency at Breakthrough in the five-spot pattern models
Figure D-4. Areal Sweep Efficiency after Breakthrough in the five-spot pattern models
45 Estimation of Recovery Factor During a Waterflood
3. Determining the vertical sweep efficiency at and after breakthrough for the layered models Using Stiles’ method
Equations (7), (8) and (9) are used in order to establish the following table:
Table D - 4. EV against WOR for the layered models
Figure E - 1. Plots of recovery efficiencies vs time for the models in serie 1
67 Estimation of Recovery Factor During a Waterflood
Figure E - 2. Plots of the field water cut vs time for the models in serie 1 Serie 2: Homogeneous models with gravity, with a quarter five-spot pattern (and with ρo = 40 lb/ft
3)
Table E - 2. Simulation results for the models in serie 2
Figure E - 7. Plots of the recovery efficiencies vs. time for the models in serie 5
Serie 6: Heterogeneous models in a 2D vertical cross-section (with ρo = 54.697 lb/ft3)
Table E - 6. Simulation results for the models in serie 6
w
o
Field Life (days)
Pore Volume (MMrb)
Recoverable Reserves (MMrb)
STOIIP (MMstb)
NP (MMstb)
Time to Breakthrough (days)
RF (simulation)
At fw = 0.95
At breakthrough
1 4100 1.603 1.218 1.16 0.645 560 0.555 0.305
2 5000 1.603 1.218 1.16 0.558 750 0.481 0.240
5 6800 1.603 1.218 1.16 0.449 1100 0.387 0.150
10 8550 1.603 1.218 1.16 0.381 1400 0.328 0.101
71 Estimation of Recovery Factor During a Waterflood
Figure E - 8. Plots of the recovery efficiencies vs. time for the models in serie 6 Serie 7: Layered models with a quarter five-spot pattern (and with ρo = 54.697 lb/ft
3)
Table E - 7. Simulation results for the models in serie 7
w
o
Field Life (days)
Pore Volume (MMrb)
Recoverable Reserves (MMrb)
STOIIP (MMstb)
NP (MMstb)
Time to Breakthrough (days)
RF (simulation)
At fw = 0.95
At breakthrough
1 12900 16.03 12.182 11.606 6.228 650 0.536 0.175
2 11350 16.03 12.182 11.606 5.383 480 0.464 0.118
5 13350 16.03 12.182 11.606 4.276 600 0.368 0.06
10 18050 16.03 12.182 11.606 3.611 800 0.311 0.04
Figure E - 9. Plots of the recovery efficiencies vs. time for the models in serie 7
72 Estimation of Recovery Factor During a Waterflood
Figure E - 10. Plots of the field water cut vs. time for the models in serie 7
Serie 8: Layered models with gravity, with a quarter five-spot pattern (and with ρo = 40 lb/ft3)
Table E - 8. Simulation results for the models in serie 8
Figure E - 15. Plots of the recovery efficiencies vs. time for the models in serie 11 Serie 12: Heterogeneous models with a quarter nine-spot pattern (and with ρo = 54.697 lb/ft
3)
Table E - 12. Simulation results for the models in serie 12
Figure E - 18. Plots of the recovery efficiencies vs. time for the models in serie 13 Serie 14: Layered models with a quarter nine-spot pattern (and with ρo = 54.697 lb/ft
3)
Table E - 14. Simulation results for the models in serie 14
w
o
Field Life (days)
Pore Volume (MMrb)
Recoverable Reserves (MMrb)
STOIIP (MMstb)
NP (MMstb)
Time to Breakthrough (days)
RF (simulation)
At fw = 0.95
At breakthrough
1 2900 16.03 12.182 11.606 5.534 107 0.477 0.120
2 3800 16.03 12.182 11.606 4.753 140 0.410 0.080
5 4750 16.03 12.182 11.606 3.573 162 0.308 0.038
10 5900 16.03 12.182 11.606 2.854 200 0.246 0.022
78 Estimation of Recovery Factor During a Waterflood
Figure E - 19. Plots of the recovery efficiencies vs. time for the models in serie 14
Figure E - 20. Plots of the field water cut vs. time for the models in serie 14
79 Estimation of Recovery Factor During a Waterflood
Serie 15: Layered models with gravity, with a quarter nine-spot pattern (and with ρo = 40 lb/ft3)
Table E - 15. Simulation results for the models in serie 15
w
o
Field Life (days)
Pore Volume (MMrb)
Recoverable Reserves (MMrb)
STOIIP (MMstb)
NP (MMstb)
Time to Breakthrough (days)
RF (simulation)
At fw = 0.95
At breakthrough
1 3200 16.03 12.182 11.605 5.828 100 0.502 0.108
2 4500 16.03 12.182 11.605 5.246 137 0.452 0.083
5 5950 16.03 12.182 11.605 4.130 154 0.356 0.040
10 6950 16.03 12.182 11.605 3.259 200 0.280 0.022
Figure E - 21. Plots of the recovery efficiencies vs. time for the models in serie 15
Serie 16: Homogeneous models with a direct line drive pattern (and with ρo = 54.697 lb/ft3)
Table E - 16. Simulation results for the models in serie 16
Direct Line Drive 3060 8.380764 6.704611 6.638393 2.42863 0.365
86 Estimation of Recovery Factor During a Waterflood
Figure E – 31. Plots of Recovery Factors vs. time for complex models with different well patterns
Table E – 23. Simulation results for the complex models without gravity (5-spot)
w
o
Field Life (days)
Pore Volume (MMrb)
Recoverable Reserves (MMrb)
STOIIP (MMstb)
NP (MMstb)
RF (simulation)
At fw = 0.95
1 6420 8.380764 6.704611 6.638393 3.78156 0.569
2 4560 8.380764 6.704611 6.638393 3.42496 0.515
10 3000 8.380764 6.704611 6.638393 2.07845 0.313
87 Estimation of Recovery Factor During a Waterflood
Figure E – 32. Plots of Recovery Factors vs. time for complex models without gravity
Table E – 24. Simulation results for the complex models with gravity (5-spot)
w
o
Field Life (days)
Pore Volume (MMrb)
Recoverable Reserves (MMrb)
STOIIP (MMstb)
NP (MMstb)
RF (simulation)
At fw = 0.95
1 6300 8.380727 6.704581 6.638323 3.77431 0.568
2 4560 8.380727 6.704581 6.638323 3.42933 0.516
88 Estimation of Recovery Factor During a Waterflood
Figure E – 33. Plots of Recovery Factors vs. time for complex models with gravity
89 Estimation of Recovery Factor During a Waterflood
APPENDIX F: Brief Summary of the Results Obtained from MODEL II The 84 models defined in MODELS II, contain a large a transition zone in their geology, calculations of the recovery factors were
made by using the same methods previously described; Stiles’ and Dykstra-Parsons. A comparison of the theoretical results with
the simulation prediction is illustrated in Figure F – 1;
Figure F – 1. Recovery Factor Comparison for the models in MODEL II
The equations of the trendlines for each method and the values of their linear correlation coefficients are very different from the
line of perfect match defined by y = x and R2 = 1. This shows that the theoretical predictions are even more inconsistent with the
simulation results for the models with transition zone, reaffirming the fact that it is very difficult to analytically predict waterflood
performance for more complex petroleum reservoirs.